Problem: What is the least positive integer with exactly five distinct positive factors?
Answer: Recall that we can determine the number of factors of $n$ by adding $1$ to each of the exponents in the prime factorization of $n$ and multiplying the results.  We work backwards to find the smallest positive integer with $5$ factors. Since 5 is prime, the only way for a positive integer to have 5 factors is for the only exponent in its prime factorization to be 4.  The smallest fourth power of a prime number is $2^4=\boxed{16}$.